Mass Flow Averaged

The mass flow averaged scalar quantity through a surface is computed as:

Figure 1. EQUATION_DISPLAY

M a s s F l o w A v e r a g e \equiv \frac{\oint ρ ϕ | v \cdot d a |}{\oint ρ | v \cdot d a |} = \frac{\sum_{f}^{} ρ_{f} ϕ_{f} | v_{f} \cdot a_{f} |}{\sum_{f}^{} ρ_{f} | v_{f} \cdot a_{f} |}

(392)

where $ϕ_{f}$ is the face value of the selected scalar, $ρ_{f}$ is the face density, $v_{f}$ is the face velocity vector, and $a_{f}$ is the face area vector.

This equation changes if the Signed Mass Flow property is activated: the absolute value signs disappear.

See also the properties of this statistical report.

Note	When the Mass Flow and Mass Flow Averaged reports are run on derived parts with moving meshes as inputs, they behave as though the derived part were instantaneously fixed to the mesh, and use the fluid velocity relative to the Region mesh to compute mass flow.

Using Signed Mass Flow

The signed mass flow allows the following relation always to be true: if you have some mass-specific quantity h carried by the flow, (total flow of h) = (mass flow average of h)*(mass flow).

To use signed mass flow, activate the Signed Mass Flow property of the report node.

If the mass flow average is computed using signed mass flow, it may be outside the range of values encompassed by the field function being averaged.

Differences between Signed and Absolute Values

When using the signed mass flow average option, you may encounter a value greater than that obtained using the unsigned mass flow average option (default).

This is due to the changed definition now taking the sum of cell scalar quantities, multiplied by the mass flow rate associated with it, over the total mass flow rate. In a section of the flow displaying a reversed flow, the numerator and denominator individually are necessarily less for the signed mass flow average report than the unsigned. There is no guarantee, however, that the quantities change such that the ratio returned is less than that of a signed mass flow average.

For example: the scalar quantity could hold a very small number within the reversed flow region and a much larger one in the regular flow region, as such the numerator only decreases by a relatively small amount while the denominator is reduced based on the reversed flow. If the reverse flow is relatively great across the boundary then the ratio is greater for a signed mass flow average than an unsigned.

Unexpectedly Large Values

In some cases the signed mass flow average report may return a value that is much larger than any value one would expect for the input scalar quantity (or in some cases just unusual or unexpected values). The unexpected values returned are typically the result of a near divide by zero when computing the report. This may occur in a situation in which there is a zero mass flow (or very near) across the input parts of the report. This behavior is not unexpected, even though the report appears technically undefined in such situations.

However, the larger value is simply the result of this technique. Recall the average is (sum of scalar times mass flow)/(sum of mass flow), where each sum is over faces. If the mass flow is taken as the signed mass flow (and there is reversed flow), the denominator is smaller than if it were taken as absolute value, but if the negative face mass flow values correspond to small values of the scalar function on those faces, the overall ratio can be larger. With the signed mass flow, the mass flow average can be outside the range of the scalar on the input parts.

Additionally, this behavior can occur in a situation using multiple parts as input to a single mass flow average report, provided that their net mass flow rate is zero.

Mass Flow Averaged in Harmonic Balance

When the Harmonic Balance model is selected, the mass flow averaged scalar quantity through a surface is computed as:

Figure 2. EQUATION_DISPLAY

R = \frac{\sum_{f} {\hat{m}}_{0 f} {\tilde{ϕ}}_{f}}{\sum_{f} {\hat{m}}_{0 f}}

(393)

where:

${\hat{m}}_{0 f}$ is the time-mean mass flow rate (or face flux) at face $f$ .
${\tilde{ϕ}}_{f} = C \cdot ϕ_{f}^{*}$ is the time-mean scalar at face $f$ for Volume Mesh Representation and the specific variant of the scalar requested by the HB Solution View for HB Solution View Representation (for example, Scalar at a specific time instant or Fourier Mode 1 Imag of Scalar).